I used to have this fear, but for me I personally thought to myself..... "Come on, no meaning to life? No bigger picture? We are simply here because two atoms collided and had a Big Bang???" I Wasn't buying it. I believe in a higher power without a doubt. Look at the sky, the stars, flowers, the feeling of love, embarrassment, laughing with others, tears, etc. science can only tell us so much. And I'll remind you the Big Bang is simply a theory. Not a proven one. I'm not pushing you to find God, I personally don't like to push religion onto people, or push them against it. If it's for you then it's for you. But it's what helped me when I had a similar fear.
This is something nobody knows and can't know. But thinking about it can be either philosophical or fun or terrifying depending on how we choose to approach it. But I would say, if it's just sleep without dreams, that would be great, so what are you afraid of?